Finite Group Databasebeta

← #13088218

Subgroups — #13088218

Order 1440.

408 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C5×#41C211yesmaximal, characteristic, complemented
2C2×C4⋊(C3⋊D15)C211yesmaximal, characteristic, complemented
3#10623911C212yesmaximal, complemented
4#10623909C212yesmaximal, complemented
5#10623904C211yesmaximal, characteristic, complemented
6#91838—34nomaximal
7C22×C3⋊D15—21no—
8C32×C5×Q8C2211yescharacteristic, complemented
9C32×C5×D4—21no—
10C32×C5×D4C2211yescharacteristic, complemented
11C2×C32×C4×C5—21no—
12C2×C32×C4×C5C2211yescharacteristic
13C4⋊(C3⋊D15)—21no—
14C4⋊(C3⋊D15)C2212yes—
15C3⋊(C3⋊#168)C2212yescomplemented
16#64158—51nomaximal, Hall
17C3×C5×#41D314yescomplemented
18C2×D60—34no—
19#6717—38no—
20#6715—38no—
21#6710—34no—
22C22×C32×C5—21no—
23C22×C32×C5D411yescharacteristic
24C2×C3⋊D15—22no—
25C2×C3⋊D15—42no—
26C32×C4×C5—21no—
27C32×C4×C5D411yescharacteristic
28C32×C4×C5C2311yescharacteristic
29#4095—91noHall
30C32×#41D511yescharacteristic, complemented
31C2×C4⋊(C3⋊D3)—51no—
32#3755—52no—
33#3753—52no—
34#3748—51no—
35C22×D15—64no—
36C3×C5×Q8C2×D314yescomplemented
37C3×C5×D4—24no—
38C3×C5×D4C2×D314yescomplemented
39C2×C3×C4×C5—24no—
40C2×C3×C4×C5C2×D314yes—
41D60—38no—
42D60—64no—
43C3⋊#168—38no—
44#949—154no—
45C2×C32×C5—21no—
46C2×C32×C5—41no—
47C2×C32×C5C2×D411yescharacteristic
48C3⋊D15—42no—
49C5×#41C3⋊D311yescharacteristic, complemented
50C2×D20—91no—
51#688—92no—
52#686—92no—
53#681—91no—
54C22×C3⋊D3—101no—
55C32×Q8C2×D511yescharacteristic, complemented
56C32×D4—21no—
57C32×D4C2×D511yescharacteristic, complemented
58C2×C32×C4—21no—
59C2×C32×C4C2×D511yescharacteristic
60C4⋊(C3⋊D3)—52no—
61C4⋊(C3⋊D3)—101no—
62C3⋊(C3⋊C8)—52no—
63C22×C3×C5—24no—
64C22×C3×C5#6714yes—
65C2×D15—68no—
66C2×D15—128no—
67C3×C4×C5—24no—
68C3×C4×C5#6714yes—
69C3×C4×C5C22×D314yes—
70C3×#41D1514yescomplemented
71C2×D12—154no—
72#215—158no—
73#213—158no—
74#208—154no—
75C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
76C22×D5—181no—
77C5×Q8C2×C3⋊D311yescharacteristic, complemented
78C5×D4—21no—
79C5×D4C2×C3⋊D311yescharacteristic, complemented
80C2×C4×C5—21no—
81C2×C4×C5C2×C3⋊D311yescharacteristic
82D20—92no—
83D20—181no—
84#168—92no—
85C22×C32—21no—
86C22×C32C22⋊D511yescharacteristic
87C2×C3⋊D3—102no—
88C2×C3⋊D3—202no—
89C32×C4—21no—
90C32×C4C22⋊D511yescharacteristic
91C32×C4C22×D511yescharacteristic
92AGL(1,ℤ/8)—451noSylow
93C2×C3×C5—24no—
94C2×C3×C5—44no—
95C2×C3×C5C2×#6714yes—
96D15—128no—
97C22×D3—304no—
98C3×Q8C2×D1514yescomplemented
99C3×D4—24no—
100C3×D4C2×D1514yescomplemented
101C2×C3×C4—24no—
102C2×C3×C4C2×D1514yes—
103D12—158no—
104D12—304no—
105C3⋊C8—158no—
106C22×C5—21no—
107C22×C5#63811yescharacteristic
108C2×D5—182no—
109C2×D5—362no—
110C4×C5—21no—
111C4×C5#63811yescharacteristic
112C4×C5C22×C3⋊D311yescharacteristic
113C2×C32—21no—
114C2×C32—41no—
115C2×C32C2×C22⋊D511yescharacteristic
116C3⋊D3—202no—
117#41C3⋊D1511yescharacteristic, complemented
118C2×D4—451no—
119SD16—452no—
120D8—452no—
121M16—451no—
122C3×C5#94914yescomplemented
123C22×C3—24no—
124C22×C3#122014yes—
125C2×D3—308no—
126C2×D3—608no—
127C3×C4—24no—
128C3×C4#122014yes—
129C3×C4C22×D1514yes—
130C2×C5—21no—
131C2×C5—41no—
132C2×C5C2×#63811yescharacteristic
133D5—362no—
134C32#409511yescharacteristic, complemented, Sylow
135C23—901no—
136Q8C2×C3⋊D1511yescharacteristic, complemented
137D4—21no—
138D4—452no—
139D4—901no—
140D4C2×C3⋊D1511yescharacteristic, complemented
141C2×C4—21no—
142C2×C4C2×C3⋊D1511yescharacteristic
143C8—452no—
144C2×C3—24no—
145C2×C3—44no—
146C2×C3C2×#122014yes—
147D3—608no—
148C5#6415811yescharacteristic, complemented, Sylow
149C22—21no—
150C22—902no—
151C22—1802no—
152C22#6718411yescharacteristic
153C4—21no—
154C4#6718411yescharacteristic
155C4C22×C3⋊D1511yescharacteristic
156C3#9183814yescomplemented
157C2—21no—
158C2—41no—
159C2—1802no—
160C2C2×#6718411yescharacteristic
subgroupG/H
centerC2C2×#67184
commutator, cosocleC32×C4×C5C23
FrattiniC4C22×C3⋊D15
FittingC32×C5×#41C2
socleC2×C32×C5C2×D4
hypercenter#41C3⋊D15
hypocenterC32×C5AGL(1,ℤ/8)
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C32×C5×#41C2semidirect
2. C2×C4⋊(C3⋊D15)C2semidirect
3. #10623911C2semidirect
4. #10623909C2semidirect
5. #10623904C2semidirect
8. C32×C5×Q8C22semidirect
10. C32×C5×D4C22semidirect
15. C3⋊(C3⋊#168)C22semidirect
17. C3×C5×#41D3semidirect
30. C32×#41D5semidirect
36. C3×C5×Q8C2×D3semidirect
38. C3×C5×D4C2×D3semidirect
49. C5×#41C3⋊D3semidirect
55. C32×Q8C2×D5semidirect
57. C32×D4C2×D5semidirect
70. C3×#41D15semidirect
75. C32×C5AGL(1,ℤ/8)semidirect
77. C5×Q8C2×C3⋊D3semidirect
79. C5×D4C2×C3⋊D3semidirect
98. C3×Q8C2×D15semidirect
100. C3×D4C2×D15semidirect
117. #41C3⋊D15semidirect
122. C3×C5#949semidirect
134. C32#4095semidirect
136. Q8C2×C3⋊D15semidirect
140. D4C2×C3⋊D15semidirect
148. C5#64158semidirect
156. C3#91838semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×C5×#412
C2×C4⋊(C3⋊D15)2
#106239112
#106239092
#106239042
#918383
#641585